Questions tagged [exponential-function]

For question involving exponential functions and questions on exponential growth or decay.

953 questions
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Bounding $n$-th derivative of $x \mapsto \exp\left({-\frac{1}{x^2}}\right)$

Let $f : x \mapsto \exp\left({-\frac{1}{x^2}}\right)$ defined for all $x \in \mathbb{R}^*$. It is quite easy to prove by induction that $f$ can be continuated in $0$ in a $\mathcal{C}^\infty$ ...
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How to calculate $e^{0.1}$ using taylor inequality

How can I calculate $e^{0.1}$ using Taylor series and Inequality. I know that the formula is $| R_n (x) | \leq \dfrac {M | x- a|^{n+1}} {(n+1)!}$ But I dont know how to apply it
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Simplifying complicated integral including CDF of normal distribution

I'm dealing with a complicated integral and want to know whether you are aware of some possibility how to simplify it. If you think, there is no possibility, please tell me. Then I'll use numerics. ...
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Non linear regression calculator

Are there any really good non linear regression calculators around the web? Or is something like matlab the best solution? I tried using excel and its solver tool, but it's complete garbage lol. ...
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How to use Taylor series to get $e^x\geq1+x$

I know that from $$e^x=\sum_{i=0}^\infty \left(\frac{x^i}{i!}\right)$$ we can get the inequality $e^x\ge1+x$. But how?
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Evaluating a difficult integral

I need to evaluate this integral $$\int_{0}^{1}\mathrm dx\, x^{m}\exp\left[-k\frac{x}{(1-x^2)^{2/3}}\right],$$ where the real number $k>0$ and $m$ is an integer, $m\geq 0$. Is there any way to get ...
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Level curves of $e^{\alpha z}$

Sketch the families of level curves of $u$ and $v$ for the function $f=u+iv$ given by $f(z)=e^{\alpha z}$, where $\alpha$ is complex. My work so far: \begin{align*} e^{\alpha z} &= e^{(a+ib)(x+iy)...
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Yacov Perelman Nepero game

This is my first question, so sorry if I'll make any mistake in using the site formatting. I found this game on a book by Yacov Perelman and I thought it could be nice to introduce Nepero number to ...
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Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
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